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Exploring Linear Relations Among Laurent Coefficients of Certain Hilbert Series
Barringer, Austin T.
Barringer, Austin T.
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URCAS, Student research, 2018 Spring, Class of 2019, Mathematics and Computer Science, Department of, Laurent series, Hilbert algebras, Gorenstein rings
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Abstract
In [Herbig-Herden-Seaton, arXiv:1605.01572 [math.CO] 2016], the authors considered rational
functions of one variable, t, that satisfy a functional equation h(t). The function is in terms of
integers a and d, where d is the pole order of h(t) at t=1. They found that depending on the value
of r, where r=-(a+d), the coefficients of the Laurent expansion at t=1 satisfy various triangular
linear relations, and so formed structures like that of Pascal's triangle or the Lucas triangle, for
example. In this project, we experimentally investigate the extension of their findings, using a
large collection of functions of two variables t1 and t2 that satisfy an analogous equation. We
explore, using series expansions on Mathematica, whether the two variable functions are
characterized by similar linear constraints (defined iteratively as the Laurent coefficients at t2 = 1 of the Laurent coefficients at t1 = 1). Our results suggest that there do seem to be similar
relations, indicating a possible generalization to this case. Beyond the scope of this project, the
motivation for studying these relations is as follows. By a theorem of R. Stanley, a graded
Cohen-Macaulay domain A, where a is the a-invariant, is Gorenstein if and only if its Hilbert
series satisfies the functional equation given above.
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Presentation by Austin Barringer ('19), delivered at the Rhodes College Undergraduate Research and Creative Activity Symposium (URCAS).